Redshift and Velocity: Deriving Expression & Determining Velocity Direction

[A_I_]

A 14.4 KeV photon from 57 Fe is red shifted as it rises from a sourceat ground level to an absorber placed at the top of a tower of a height of 20 m because it has to expend energy to climb the gravitational potential. Derive an expression for thered shift as a fraction of the energy of the photon. What velocity of the absorber foil would be needed to compensate redshift and in which direction?


so first i found the formula f/fo = sqrt(1-b/1+b)
and to find the velocity v = bc where c is the speed of light.
so basically the natural frequency fo is related to the energy of the photon which is 14.4 KeV.
But I need to find the other frequency in order to solve for Beta and thus find the velocity. I do NOT know how to do this.
There must be a formula relating the height to the enrgy or to the frequency.
And for the second part i said: since it is redshifted we have an increase in the wavelength thus a decrease in energy and decrease in velocity.
So it is in the opposite direction of the source.
Is it right?


It's urgent :)
Thanks for any help :)
Joe

 

[Andrew Mason]

A 14.4 KeV photon from 57 Fe is red shifted as it rises from a sourceat ground level to an absorber placed at the top of a tower of a height of 20 m because it has to expend energy to climb the gravitational potential. Derive an expression for thered shift as a fraction of the energy of the photon. What velocity of the absorber foil would be needed to compensate redshift and in which direction?

so first i found the formula f/fo = sqrt(1-b/1+b)
and to find the velocity v = bc where c is the speed of light.
so basically the natural frequency fo is related to the energy of the photon which is 14.4 KeV.
But I need to find the other frequency in order to solve for Beta and thus find the velocity. I do NOT know how to do this.
There must be a formula relating the height to the enrgy or to the frequency.
And for the second part i said: since it is redshifted we have an increase in the wavelength thus a decrease in energy and decrease in velocity.
So it is in the opposite direction of the source.
Is it right?

Gravitational redshift is best explained using energy. The energy of the photon is: $E = mc^2$. The m is the photon's relativistic mass: $m=E/c^2$. The gain in energy, $\Delta E = mgh$. Use the expression for relativistic mass to substitute for m.

That gives you the change in energy. How would you determine the change in frequency or wavelength?

Use your relativistic doppler approach to find the speed of the absorber that would compensate for the gravitational redshift.

Now a photon traveling upward in a gravitational field is equivalent to a photon traveling in 0 gravity viewed by an observer moving with acceleration = g (the Principle of Equivalence). So you can think of the gravitational redshift of a photon in moving from ground to a height h as equivalent to doppler shift from an observer moving with acceleration = g.

If the photon is emitted when its speed is 0 and absorbed when its speed is v = at, where t is the time it takes for the photon to go from ground level to height h, the observer will observe a doppler shift which should equal to the gravitational redshift observed by the stationary observer in the gravitational field.

Work out the expression for doppler shift at the absorber in terms of the v of this moving observer absorbing the photon at that point. Check to see if it is the same as the gravitational redshift that you worked out using the energy approach.

AM